Lanczos thought that this was cumbersome, and preferred the more compact antiderivative formulation (see [1]} pp 125-126)
\(\int T_k(x)dx=\frac{1}{2(k+1)} T_{k+1}(x)-\frac{1}{2(k-1)}T_{k-1}(x)+\frac{k\sin {k\pi /2}}{k^2-1}\:,\) 
